What is the sum of an even integer and an odd integer?

The sum of an even integer and an odd integer is always an odd integer. This is a fundamental property in number theory that holds true for any combination of an even and an odd number.

Understanding Even and Odd Integers

To fully grasp why their sum is odd, it's helpful to first define what constitutes an even or an odd integer:

• Even Integer: An integer that is perfectly divisible by 2, leaving no remainder. It can be expressed in the form $2k$, where $k$ is any integer.
  • Examples: ..., -4, -2, 0, 2, 4, 6, ...
• Odd Integer: An integer that is not perfectly divisible by 2. It can be expressed in the form $2k + 1$, where $k$ is any integer.
  • Examples: ..., -3, -1, 1, 3, 5, 7, ...

The Sum: An Odd Result

As explicitly stated by the Art of Problem Solving, "The sum of an even integer and an odd integer is odd.". This principle is consistent and can be mathematically proven.

Why is the sum always odd? (Mathematical Insight)

Let's use the algebraic definitions of even and odd integers to illustrate this property:

1. Represent an Even Integer (E): An even integer can be written as $E = 2k$, where $k$ is any integer.
2. Represent an Odd Integer (O): An odd integer can be written as $O = 2m + 1$, where $m$ is any integer.
3. Calculate Their Sum:
   $E + O = (2k) + (2m + 1)$
   $E + O = 2k + 2m + 1$
   $E + O = 2(k + m) + 1$

Since $k$ and $m$ are both integers, their sum $(k + m)$ will also be an integer. Let's call this new integer $J = (k + m)$.

Therefore, the sum simplifies to $2J + 1$. This form, $2 \times (\text{an integer}) + 1$, is precisely the definition of an odd integer. This algebraic proof confirms that the sum of an even integer and an odd integer will always yield an odd result.

Practical Examples

Here are a few numerical examples demonstrating this rule:

• $2 \text{ (even)} + 3 \text{ (odd)} = 5 \text{ (odd)}$
• $10 \text{ (even)} + 7 \text{ (odd)} = 17 \text{ (odd)}$
• $-4 \text{ (even)} + 1 \text{ (odd)} = -3 \text{ (odd)}$
• $0 \text{ (even)} + 9 \text{ (odd)} = 9 \text{ (odd)}$

Summary of Integer Sum Properties

Understanding how even and odd integers interact through addition can be summarized in the following table, highlighting all possible combinations:

This table clearly illustrates that the only combinations resulting in an odd sum are when one integer is even and the other is odd.